Theory of x-ray scattering from optically pumped excitons in atomically thin semiconductors

Abstract

We propose a framework to explore the internal charge distribution of mesoscopic quasiparticles by inelastic x-ray scattering, while also accounting for the conventional scattering from electrons. Specifically, we investigate a new contribution of intrinsic and optically pumped excitons (bound electron-hole pairs) to the x-ray scattering spectrum of transition metal dichalcogenides (TMDCs). The optical excitation leads to the creation of Wannier exciton populations, adding new quasi-elastic processes beyond the conventional electronic features to the x-ray scattering spectra. Differential spectra (with and without optical pumping) can be used to isolate and identify the internal charge distribution of the optically pumped excitons in the scattering response, potentially offering insights into many-body interactions and quasi-particle dynamics in 2D systems.

Figures (9)

• Figure 1: x-ray scattering geometry: A free-standing atomically thin semiconductor is optically excited by a laser with vector potential $\boldsymbol{A}_{o}$ at a frequency resonant with the band gap of the material, creating excitons. The incoming x-ray field with vector potential $\boldsymbol{A}_{X}$ of wavevector $\boldsymbol{k}_{X}$ and frequency $\omega_{X}$ is scattered by the excited electronic system. The scattered field is frequcency resolved at different wavevectors $\boldsymbol{k}$, giving the spectrum $S(\Delta\boldsymbol{q},\omega)$ at a particular momentum transfer $\Delta\boldsymbol{q}=\boldsymbol{k}_{X}-\boldsymbol{k}$.
• Figure 2: Electron-hole picture vs. exciton picture. The energy of the band gap is shown by a vertical dashed line in both pictures. Left: Bandstructure of a two-band semiconductor in a parabolic approximation with dispersion $\epsilon^{c/v}_{\boldsymbol{k}}$ of the conduction/valence band. An electron with momentum $\boldsymbol{k_1}$ is created in the conduction band and a hole with momentum $\boldsymbol{k_2}$ is created in the valence band. This transition can be described by the pair-operator $P^{\dagger}_{\boldsymbol{k_1},\boldsymbol{k_2}}$. Right: A Coulomb-bound electron-hole pair, an exciton, is created with center-of-mass momentum $\boldsymbol{Q}=\boldsymbol{k_1}-\boldsymbol{k_2}$ and quantum number $\nu$. This can be described by the exciton operator $P^{\dagger\, \nu}_{\boldsymbol{Q}}$. The exciton dispersion is given by $\epsilon^{\nu}_{\boldsymbol{Q}}$. Dispersions of other excitonic quantum numbers are indicated as dashed lines. Above the band gap energy the continuum is visualized by the hatched region.
• Figure 3: Intrinsic x-ray scattering spectra for the material WS$_2$ (without optical pumping). Due to reasons of visibility the elastic scattering contribution was normalized with respect to the highest value of the excitonic contribution and we used also the same linewidth as for the excitons. The spectrum is normalized with respect to the maximum value. (a) Energy loss and momentum transfer dependent spectrum. The vertical white dashed line denotes the band gap energy while the horizontally dashed lines are cuts of the spectrum which are shown in (b). To improve comparability, the colors used in (b) match the colors of the cuts (horizontal dashed lines).
• Figure 4: Full spectrum $S(\Delta\boldsymbol{q},\omega)$ of WS$_2$ including elastic and inelastic contributions from the pumped semiconductor. The total spectrum is normalized. As in Fig. \ref{['fig:spectrum_unpumped_for_WS2']} the contribution from the elastic scattering was scaled to the highest value of the excitonic contribution. (a) Energy loss and momentum transfer dependent spectrum. The vertical white dashed line shows the energy of the band gapwhile the horizontally dashed lines are cuts of the spectrum shown in (b). For a better visibility the colors of the dashed lines are matching with the colors used in (b).
• Figure 5: Differential scattering spectrum of WS$_2$ with contributions from 1s-excitons only. The spectrum is normalized with respect to the maximum value. In the inset we see the absolute square of the convolution of the 1s-exciton wavefunctions, which could be detected by integrating the spectrum with respect to the energy $\hbar\omega$.